HIGHER-DIMENSIONAL SHRINKING TARGET PROBLEM FOR BETA DYNAMICAL SYSTEMS

We consider the two-dimensional shrinking target problem in beta dynamical systems (for general beta > 1) with general errors of approximation. Let f, g be two positive continuous functions. For any x(0), y(0) is an element of [0, 1], define the shrinking target set E(T-beta, f, g) := {(x, y) is an element of [0, 1](2): vertical bar T(beta)(n)x - x(0)vertical bar < e(-Snf(x)) vertical bar T(beta)(n)y - y(0)vertical bar < e(-Sng(y)) for infinitely many n is an element of N}, where S(n)f (x) = Sigma(n-1)(j=0) f(T(beta)(j)x) is the Birkhoff sum. We calculate the Hausdorff dimension of this set and prove that it is the solution to some pressure function. This represents the first result of this kind for the higher-dimensional beta dynamical systems.